Mathieu transformation

The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form

${\displaystyle \sum _{i}p_{i}\delta q_{i}=\sum _{i}P_{i}\delta Q_{i}\,}$

The transformation is named after the French mathematician Émile Léonard Mathieu.

In order to have this invariance, there should exist at least one relation between ${\displaystyle q_{i}}$ and ${\displaystyle Q_{i}}$ only (without any ${\displaystyle p_{i},P_{i}}$ involved).

${\displaystyle {\begin{aligned}\Omega _{1}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&=0\\&{}\ \ \vdots \\\Omega _{m}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&=0\end{aligned}}}$

where ${\displaystyle 1<m\leq n}$. When ${\displaystyle m=n}$ a Mathieu transformation becomes a Lagrange point transformation.